Discrete Space is Locally Path-Connected
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Theorem
Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is locally path-connected.
Proof
From Set in Discrete Topology is Clopen, $\set a$ is open in $T$.
From Basis for Discrete Topology, the set:
- $\BB := \set {\set x: x \in S}$
is a basis for $T$.
Let $\set x \in \BB$.
From Point is Path-Connected to Itself, it follows that $\set x$ is path-connected.
Hence $T$ has a basis consisting entirely of path-connected sets.
So by definition $T$ is locally path-connected.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $10$